Advances in Differential Equations

Semiclassical limit for a quasilinear elliptic field equation: one-peak and multipeak solutions

Marino Badiale, Vieri Benci, and Teresa D'Aprile

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This paper deals with the existence of one-bump and multibump solutions for the following nonlinear field equation: $$-\Delta u+V(h x)u-\Delta_{p}u+ W'(u)=0$$ where $u:\mathbb R^{N}\rightarrow\mathbb R^{N+1},$ $N\geq 2,$ $p>N,$ $h>0,$ the potential $V$ is positive and $W$ is an appropriate singular function. Existence results are established provided that $h$ is sufficiently small, and we find solutions exhibiting a concentration behaviour in the semiclassical limit (i.e., as $h\rightarrow 0^{+}$) at any prescribed finite set of local minima, possibly degenerate, of the potential. Such solutions are obtained as local minima for the associated energy functional. No restriction on the global behaviour of $V$ is required except that it is bounded below away from zero. In the proofs of these results we use a variational approach, and the method relies on the study of the behaviour of sequences with bounded energy, in the spirit of the concentration-compactness principle.

Article information

Adv. Differential Equations Volume 6, Number 4 (2001), 385-418.

First available in Project Euclid: 2 January 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J50: Variational methods for elliptic systems
Secondary: 35A05 35B05: Oscillation, zeros of solutions, mean value theorems, etc.


Badiale, Marino; Benci, Vieri; D'Aprile, Teresa. Semiclassical limit for a quasilinear elliptic field equation: one-peak and multipeak solutions. Adv. Differential Equations 6 (2001), no. 4, 385--418.

Export citation